Of course good zombie movies exist. If you want some good ones, check out Romero's Living Dead trilogy (Night of the Living Dead, Dawn of the Dead, Day of the Dead), Lucio Fulci's Zombie, Fulci's City of the Living Dead, and Jorge Grau's Let Sleeping Corpses Lie (aka Don't Open the Window, aka Breakfast at the Manchester Morgue). Some might also include Fulci's The Beyond with this list, but I don't really consider it to be a zombie movie, even though there are zombies in it. I'd also recommended Tombs of the Blind Dead and Return of the Blind, but again I don't really consider them zombie movies per say.

And just for the hell of it, I'm going to list some good-bad ones as well: Umberto Lenzi's Nightmare City (aka City of the Walking Dead), Bruno Mattei's Hell of the Living Dead (aka Night of the Zombies), and even Fulci's/Mattei's Zombi 3. Oh yeah, and don't let me forget Burial Ground: Nights of Terror!